Consider Eq. (14.17) with \(l=0\) and \(V(r)=-V_{0} e^{-\frac{r}{a}}\). Change variables from \(r\) to \(z=e^{-\frac{r}{2 a}}\), and show that Bessel's equation results. What boundary conditions are to be imposed on \(\chi\) as a function of \(z\), and how can these be used to determine the energy levels? What is the lower limit to \(V_{0}\) for which a bound state exists?

The Schrödinger equation is fundamental in quantum mechanics. It describes how the quantum state of a physical system changes over time. For a single, non-relativistic particle, it is given by:
\[ -\frac{\hbar^2}{2m} abla^2 \psi + V \psi = E \psi \]
Here:

• \(\hbar\) is the reduced Planck's constant
• m is the mass of the particle
• \(abla^2\) is the Laplacian
• V is the potential energy
• \(\psi\) is the wave function
• E is the energy of the system

In our exercise, turning the potential \( V(r) = -V_0 e^{-r/a} \) into an exponential variable \( z = e^{-r/(2a)} \) simplifies solving the equation.
By substituting \( z \), we change the variable dependence and can later derive a form of Bessel's equation.

Bessel's equation is a type of differential equation that frequently arises in problems with circular or cylindrical symmetry. In our problem, the transformed Schrödinger equation takes the form of Bessel's equation:
\[ z^2 \frac{d^2 \chi}{dz^2} + z \frac{d \chi}{dz} + (k^2 z^2 - u^2) \chi = 0 \]
Here:

• z is the independent variable derived from \( r \)
• \(\chi(z)\) is the transformed wave function
• \(k\) and \(u\) are constants related to energy and potential parameters

Solving this equation yields Bessel functions of the first and second kinds. These solutions are essential in determining the behavior of \(\chi(z)\) under quantum mechanics.

Boundary conditions are crucial in solving differential equations like the Schrödinger equation. They define the behavior of the wave function at specific points and ensure physical relevance. For \(\chi(z)\), boundary conditions are derived from:

• The behavior at infinity: as \(z \rightarrow 0\)
• The behavior at zero: as \(z \rightarrow 1\)

These conditions help in finding the correct Bessel function solutions, ensuring that the wave function \(\chi(z)\) is normalizable and physically meaningful.
Applying these boundary conditions allows us to determine discrete energy levels by requiring that the wave function vanishes or reaches a finite value at the boundaries. This quantization leads to specific allowed energy values.
Analyzing these boundary conditions also helps find the lower limit of \(V_0\) needed for a bound state to exist. It ensures that the potential well is deep enough to support at least one bound state, which corresponds to a specific threshold value of \(V_0\).